By A. A. Ilyushin and V. S. Lensky (Auth.)
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Additional info for Strength of Materials
The quantities eu form a strain tensor and are called strain components. The reason why half, and not whole, shear strains occur in the strain tensor, whereas in the stress tensor we have the full value for the shear stresses, can be explained as follows. The stresses axy and oyx are equal and each acts on an area perpendicular to the axes Ox and Oy\ at the same time, yxy is a change in angle due to a combined displacement of fibres directed along the axes Ox and Oy. The region of the point O is subjected by the deformation to a rotation as a whole.
We shall assume that σ, > σ2 (this is of no significance) and we mark off on the axis of abscissae the points A and B corresponding to these values (Fig. 17). ΐ A FIG. 17. With the point 0\ the mid-point of AB (its abscissa OO' is (σί + σ2)/2), as centre, we describe a circle of radius (σχ — σ 2 )/2. This circle will pass through the points A and B. We draw the diameter CD inclined at some angle 2 E. it occurs between fibres forming angles of 45° to the principal axes of strain. e. 29) ^yy We see that in order to describe the deformed state in the vicinity of some point we can make use of a geometrical figure—a Mohr's circle diagram, with relative extensions as abscissae and half the shearing strains as ordinates. e. 30) so that the octahedral shearing strain yv is 7, = 2rçv = i]/(y\2 + yL + rli) = f W*l = l|/[(^ - £ 2) 2 + (*2 - ^3)2 + (*3 - ^l) 2 ] Cyy)2 + ~ *zzY fey + fez - £ * * ) 2 + 6 f e z + <£, + 6 * , ) ] .
E. it occurs between fibres forming angles of 45° to the principal axes of strain. e. 29) ^yy We see that in order to describe the deformed state in the vicinity of some point we can make use of a geometrical figure—a Mohr's circle diagram, with relative extensions as abscissae and half the shearing strains as ordinates. e. 30) so that the octahedral shearing strain yv is 7, = 2rçv = i]/(y\2 + yL + rli) = f W*l = l|/[(^ - £ 2) 2 + (*2 - ^3)2 + (*3 - ^l) 2 ] Cyy)2 + ~ *zzY fey + fez - £ * * ) 2 + 6 f e z + <£, + 6 * , ) ] .